Find La Placian of a function in cartesian and Spherical Coordinates

[lonewolf219]

Homework Statement

Prove the La Placian of V(x,y,z)=(zx$^{2}$)/(x$^{2}$+y$^{2}$+z$^{2}$) in Cartesian coordinates is equal to that in Spherical coordinates

Homework Equations

$\nabla$$^{2}$V=0

The Attempt at a Solution

I have attempted to calculate all the terms out, and there were A LOT. I was hoping the derivatives in Cartesian, which I did first, would cancel, but they didn't. I may have made a mistake, I used the product rule and came up with 6 terms in the numerator over (x$^{2}$+y$^{2}$+z$^{2}$)$^{3}$. Any suggestions? Spherical was even more complicated... I had the following:
r(cosθ)$^{2}$(sin$\phi$)$^{2}$(cos$\phi$) before I began taking partial derivatives. Any help would really be appreciated, thanks...

 

[vela]

Which convention for spherical coordinates are you using? Physicists typically use $\theta$ as the angle from the z-axis whereas mathematicians use $\phi$. Your expression for $V(r,\theta,\phi)$ appears to be using the math convention. I just ask because you posted this in the physics section.

No suggestions, by the way. I think you just have to grind it out.

 

[lonewolf219]

This problem is from a physics class, and the class doesn't have a book assigned...so I've been looking through my calc iv book to try and get some information. Thanks for pointing out there are different systems, I wasn't aware of that and my professor didn't mention it...